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In this paper we first establish a new critical point theorem which may produce a relationship between the sign-changing critical point, the Morse index, the upper bound of the critical value, the value of the type inf k ≤ dim ⁡ Y ∞ Y ⊂ E , ⁡ sup Y ⁡ G and the number of nodal domains. An unified approach is introduced for these composite properties. As one application of the abstract theorem, we study the existence of infinitely many sign-changing bound states for the following nonlinear stationary Schrodinger equation: − Δ u + a ( x ) u = | u | p − 2 u , u ∈ H 1 ( R N ) , where N ≥ 2 , p > 2 and p 2 N / ( N − 2 ) : = 2 ⁎ when N > 2 ; the potential a ( x ) ∈ C ( R N , R ) is a nonnegative function verifying suitable decay assumptions, but being not symmetric or periodic. We obtain the information on the precise estimates of the energies, the Morse indices and the number of nodal domains for these sign-changing bound states. We believe that the abstract theorem in this article will have more applications.

In this paper we study the nonhomogeneous semilinear fractional Schrodinger equation with critical growth $$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^s}u + u = {u^{2_s^* - 1}} + \lambda (f(x,u)) + h(x)),}&{x \in {\mathbb{R}^N},} \\ {u \in {H^s}({\mathbb{R}^N}),\;\;\;u(x) > 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in {\mathbb{R}^N},} \end{array}} \right.$$ where s ∈ (0, 1), N > 4s, and λ > 0 is a parameter, $$2_s^* = {{2N} \over {N - 2s}}$$ is the fractional critical Sobolev exponent, f and h are some given functions. We show that there exists 0 λ*, a unique solution (λ*,uλ*) if λ = λ*, which shows that (λ*,uλ*) is a turning point in HS(ℝN) for the problem. Our proofs are based on the variational methods and the principle of concentration-compactness.

In this paper, we investigate the qualitative properties of positive solutions for the following two-coupled elliptic system in the punctured space: $$ \begin{cases} -\Delta u =\mu_1 u^{2q+1} + \beta u^q v^{q+1} \\ -\Delta v =\mu_2 v^{2q+1} + \beta v^q u^{q+1} \end{cases} \textmd{in} ~\mathbb{R}^n \backslash \{0\}, $$ where $\mu_1, \mu_2$ and $\beta$ are all positive constants, $n\geq 3$. We establish a monotonicity formula that completely characterizes the singularity of positive solutions. We prove a sharp global estimate for both components of positive solutions. We also prove the nonexistence of positive semi-singular solutions, which means that one component is bounded near the singularity and the other component is unbounded near the singularity., Comment: 27 pages

We consider the following singularly perturbed nonlinear elliptic problem $$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u=f(u),\ u\in H^1({\mathbb {R}}^N), \quad N\ge 3, \end{aligned}$$ where f is of critical growth. Using the variational techniques, we construct a solution $$u_\varepsilon $$ which concentrates around the saddle points of V as $$\varepsilon \rightarrow 0$$ .

By using variational methods and the generalized Nehari manifold due to Szulkin and Weth, the existence of the ground states and the multiplicity of solutions for a wide class of superlinear elliptic equations is studied.

We study the following linearly coupled Schrodinger equations: { − e 2 Δ u + a ( x ) u = u p + λ v , x ∈ R N , − e 2 Δ v + b ( x ) v = v 2 ⁎ − 1 + λ u , x ∈ R N , u , v > 0 in R N , u ( x ) , v ( x ) → 0 as | x | → ∞ , where N ⩾ 3 , 2 ⁎ = 2 N N − 2 , 1 p 2 ⁎ − 1 , and a ( x ) , b ( x ) are positive continuous potentials which are both bounded away from 0. Under some assumptions on a ( x ) and λ > 0 , we obtain positive solutions of the coupled system for sufficiently small e > 0 , which have concentration phenomenon as e → 0 . It is interesting that we do not need any further assumptions on b ( x ) .

We are interested in two coupled nonlinear Schrodinger equations on a smooth compact boundaryless Riemannian manifold, which arises from Bose-Einstein condensates and nonlinear optics theory. The existence of least energy solutions, multiplicity of positive solutions, as well as synchronized vector solutions is obtained.We are interested in two coupled nonlinear Schrodinger equations on a smooth compact boundaryless Riemannian manifold, which arises from Bose-Einstein condensates and nonlinear optics theory. The existence of least energy solutions, multiplicity of positive solutions, as well as synchronized vector solutions is obtained.

In this paper we study the existence, multiplicity and concentration behaviour of ground states for a class of quasilinear Schrodinger equations with critical growth. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. We relate the number of positive solutions with the topology of the set where the potential attains its minimum value. The proofs are based on the Ljusternik–Schnirelmann theory and variational methods.

We study the following system of nonlinear Schrodinger equations: $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\Delta u +a(x) u = f(u)+\lambda v, \quad x\in \mathbb R ^N, \\ -\varepsilon ^2\Delta v +b(x) v =g(v)+\lambda u, \quad x\in \mathbb R ^N,\\ u,v >0 \,\,\,\hbox {in}\,\,\,\mathbb R ^N,\quad u, v \in H^1 (\mathbb R ^N), \end{array}\right. \end{aligned}$$ where \(N\ge 3\), \(\varepsilon , \lambda >0\), and \(a, b, f, g\) are continuous functions. Under very general assumptions on both the potentials \(a, b\) and the nonlinearities \(f, g\), for small \(\lambda >0\) and \(\varepsilon >0\), we obtain positive solutions of this coupled system via pure variational methods. The asymptotic behaviors of these solutions are also studied either as \(\varepsilon \rightarrow 0\) or as \(\lambda \rightarrow 0\).

We consider the existence of the ground states solutions to the following Schrodinger equation: $$- \Delta u + V(x)u = f(u), u \in H^1 \left( {\mathbb{R}^N } \right),$$ where N ⩾ 3 and f has critical growth. We generalize an earlier theorem due to Berestycki and Lions about the subcritical case to the current critical case.

In this paper we consider the nonlinear fractional Schrodinger equations in presence of a critical power nonlinearity and a subcritical term $$\begin{aligned} \varepsilon ^{2s}(-\Delta )^s u+V(x)u=f(u)+u^{2^*_s-1}\quad x\in \mathbb R^N, \end{aligned}$$ where $$s\in (0,1), 2^*_s=2N/(N-2s), N>2s, (-\Delta )^s$$ is the fractional laplacian and f(u) is of subcritical nonlinearity, and $$\varepsilon $$ is a positive parameter. Under a local condition imposed on the potential V, we relate the number of positive solutions with the topology of the set where the potential attains its minimum. In the proofs we apply variational methods, penalization techniques and Ljusternik–Schnirelmann theory.

We study the following nonlinear Schrodinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1 u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in \Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1} u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in $\om$},\quad u=v=0 \,\,\hbox{on $\partial\om$}.{cases}{displaymath} Here $\om\subset \R^N$ is a smooth bounded domain, $2^\ast:=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-\la_1(\om) 0$ and $\beta\neq 0$, where $\lambda_1(\om)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. When $\bb=0$, this is just the well-known Brezis-Nirenberg problem. The special case N=4 was studied by the authors in (Arch. Ration. Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher dimensional case $N\ge 5$}. It is interesting that we can prove the existence of a positive least energy solution $(u_\bb, v_\bb)$ {\it for any $\beta\neq 0$} (which can not hold in the special case N=4). We also study the limit behavior of $(u_\bb, v_\bb)$ as $\beta\to -\infty$ and phase separation is expected. In particular, $u_\bb-v_\bb$ will converge to {\it sign-changing solutions} of the Brezis-Nirenberg problem, provided $N\ge 6$. In case $\la_1=\la_2$, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N=4.

We study the following system of nonlinear Schrodinger equations: { − Δ u + μ u = | u | p − 1 u + λ v , x ∈ R N , − Δ v + ν v = | v | 2 ⁎ − 2 v + λ u , x ∈ R N , where N ⩾ 3 , 2 ⁎ = 2 N N − 2 , 1 p 2 ⁎ − 1 and μ , ν , λ are positive parameters satisfying 0 λ μ ν . We show that, there is some critical value μ 0 ∈ ( 0 , 1 ) , such that this system has a positive ground state solution if 0 μ ⩽ μ 0 . In the case μ > μ 0 , there exists λ μ , ν ∈ [ ( μ − μ 0 ) ν , μ ν ) such that, this system has no ground state solutions if λ λ μ , ν ; while this system has a positive ground state solution if λ > λ μ , ν . In particular, if p = 2 ⁎ − 1 , the system has no nontrivial solutions. Some further properties of the ground state solutions are also studied. This seems to be the first result for such a critical Schrodinger system.

In this paper, we consider the critical quasilinear Schrodinger equations of the form $$ -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N, $$ where N ≥ 3, 2* := 2N/(N − 2) and g(u) is of subcritical growth. We prove the existence of positive bound states which concentrate around a local minimum point of V as e → 0+.

In this paper, the relationships between sign-changing critical point theorems and the linking type theorems of M. Schechter and the saddle point theorems of P. Rabinowitz are established. The abstract results are applied to the study of the existence of sign-changing solutions for the nonlinear Schrodinger equation - Δu+V(x)u = f(x,u),u ∈ H 1 (R N ), where f(x, u) is a Caratheodory function. Problems of jumping or oscillating nonlinearities and of double resonance are considered.

Some parameter-depending linking theorems are established, which allow to produce a bounded and sign-changing Palais–Smale sequence. For even functionals, a parameter-depending fountain theorem is obtained which provides infinitely many bounded and sign-changing Palais–Smale sequences. A variant mountain pass theorem is built in cones which yields bounded, positive and negative Palais–Smale sequences. The usual Palais–Smale type compactness condition and its variants are completely not necessary for these theories. More exact locations of the critical sequences can be determined. The abstract results are applied to the Schrodinger equation with (or without) critical Sobolev exponents: − Δ u + V ( x ) u = β | u | 2 * − 2 u + f ( x , u ) , x ∈ R N , β ⩾ 0 , where 2 * is the critical Sobolev exponent. The existence of (multiple) sign-changing solutions is obtained. The positive and negative solutions are also gained as by-products. We will also show that this Schrodinger problem with jumping nonlinearity is independent of the Fucik spectrum.

The paper collects some recent results in weak linking theorems. Some applications are given, mainly in Hamiltonian systems and Schrodinger equations. A relationship between linking and sign-changing critical point is given. It is shown that the existence of solutions to elliptic equations with jumping nonlinearities is independent of the Fucik spectrum.

In this paper, we study a nonlinear Schrodinger equation with periodic potential. We assume that zero is an end point of the continuous spectrum of the Schrodinger operator. We establish some existence results of the homoclinic orbits for weak superlinear cases. To this purpose, we develop new linking theorems in Banach Spaces which provide bounded Palais-Smale sequences.

In this paper we establish some variant fountain theorems without (P.S.)-type assumption. The abstract results will be used to study the symmetric nonlinear Schrodinger equations and Dirichlet boundary value problems. Under no Ambrosetti–Rabinowitz's superquadraticity condition, we obtain infinitely many large energy and small negative energy solutions respectively.

We study the existence of semi-classical bound states of the nonlinear Schrödinger equation\begin{linenomath}$$ -\varepsilon^2\Delta u+V(x)u=f(u),\quad x\in {\bf R}^N,$$\end{linenomath}whereN≥ 3;, ϵ is a positive parameter;V:RN→ [0, ∞) satisfies some suitable assumptions. We study two cases: iffis asymptotically linear, i.e., if lim|t| → ∞f(t)/t=constant, then we get positive solutions. Iffis superlinear andf(u)=|u|p−2u+|u|q−2u, 2* >p>q> 2, we obtain the existence of multiple sign-changing semi-classical bound states with information on the estimates of the energies, the Morse indices and the number of nodal domains. For this purpose, we establish a mountain cliff theorem without the compactness condition and apply a new sign-changing critical point theorem.

We study the following coupled Schr\"{o}dinger equations which have appeared as several models from mathematical physics: \begin{displaymath} \begin{cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega,\\ -\Delta u_2 +\la_2 u_2 =\mu_2 u_2^3+\beta u_1^2 u_2, \quad x\in \Om,\\ u_1=u_2=0 \,\,\,\hbox{on \,$\partial\Om$}.\end{cases}\end{displaymath} Here $\Om\subset\RN (N=2, 3)$ is a smooth bounded domain, $\la_1, \la_2$, $\mu_1, \mu_2$ are all positive constants. We show that, for each $k\in\mathbb{N}$ there exists $\bb_k>0$ such that this system has at least $k$ sign-changing solutions (i.e., both two components change sign) and $k$ semi-nodal solutions (i.e., one component changes sign and the other one is positive) for each fixed $\bb\in (0, \bb_k)$., Comment: This work continues the study of arXiv:1212.3773. Any comment is welcome

We consider the following singularly perturbed nonlinear elliptic problem: $$-\e^2\Delta u+V(x)u=f(u),\ u\in H^1(\mathbb{R^N}),$$ where $N\ge 3$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a solution $u_\e$ of the above problem which concentrates at an isolated component of positive local minimum points of $V$ as $\e\to 0$ under certain conditions on $f$. Our result completes the study made in some very recent works in the sense that, in those papers only the subcritical growth was considered, Comment: 24 pages, 0 figures

We consider the Newtonian system − q ¨ + B ( t ) q = W q ( q , t ) -\ddot q+B(t)q = W_q(q,t) with B B , W W periodic in t t , B B positive definite, and show that for each isolated homoclinic solution q 0 q_0 having a nontrivial critical group (in the sense of Morse theory), multibump solutions (with 2 ≤ k ≤ ∞ 2\le k\le \infty bumps) can be constructed by gluing translates of q 0 q_0 . Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schrödinger equation − Δ u + V ( x ) u = g ( x , u ) -\Delta u+V(x)u = g(x,u) in R N \mathbb {R}^N , where V V , g g are periodic in x 1 , … , x N x_1,\ldots ,x_N , σ ( − Δ + V ) ⊂ ( 0 , ∞ ) \sigma (-\Delta +V)\subset (0,\infty ) , and we show that similar results hold in this case as well. In particular, if g ( x , u ) = | u | 2 ∗ − 2 u g(x,u)=|u|^{2^*-2}u , N ≥ 4 N\ge 4 and V V changes sign, then there exists a solution minimizing the associated functional on the Nehari manifold. This solution gives rise to multibumps if it is isolated.

We study the following singularly perturbed problem for a coupled nonlinear Schr\"{o}dinger system: {displaymath} {cases}-\e^2\Delta u +a(x) u = \mu_1 u^3+\beta uv^2, \quad x\in \R^3, -\e^2\Delta v +b(x) v =\mu_2 v^3+\beta vu^2, \quad x\in \R^3, u> 0, v> 0 \,\,\hbox{in $\R^3$}, u(x), v(x)\to 0 \,\,\hbox{as $|x|\to \iy$}.{cases}{displaymath} Here, $a, b$ are nonnegative continuous potentials, and $\mu_1,\mu_2>0$. We consider the case where the coupling constant $\beta>0$ is relatively large. Then for sufficiently small $\e>0$, we obtain positive solutions of this system which concentrate around local minima of the potentials as $\e\to 0$. The novelty is that the potentials $a$ and $b$ may vanish at someplace and decay to 0 at infinity., Comment: Final version, published in JMP

In this paper, we study the existence and concentration behavior of ground state solutions for a class of Schrodinger-Poisson equation with a parameter ɛ > 0. Under some suitable conditions on the nonlinearity f and the potential V, we prove that for e small, the equation has a ground state solution concentrating around global minimum of the potential V in the semi-classical limit. Also, the exponential decay of the ground state solutions is studied.

We study the existence of homoclinic orbits for first order time-dependent Hamiltonian systems z= J Hz(z, t), where H(z, t) depends periodically on t and Hz(z, t) is asymptotically linear in z as |z|→∞. We also consider an asymptotically linear Schrodinger equation in RN.